Amparo Gil Gómez, José Javier Segura Sala: "Computation of zeros and turning points of special functions"
Madrid, Spain, currently Kassel, Germany

A globally convergent fixed point method can be defined for functions satisfying second order ODEs yn''+B(x)yn'+Anyn=0 (being n the order) and verifying linear formulas, with continuous coefficients, relating its derivative with the same function and a function of a contiguous order. Theses conditions are met by a broad family of special functions (solutions of Bessel, Coulomb, Hermite, Legendre and Laguerre equations, among others) and leads to efficient algorithms for the computation of the zeros of general solutions of these equations. We show that similar techniques can be applied for the computation of turning points. The interlacing between the zeros of a solution of a second order ODE yn''+Anyn=0 and its derivatives when An>0 (at most, there can be only a zero and there can be two turning points when An0) and the monotonicity of the logarithmic derivative yn'/yn is the starting point in this case. Computer algebra systems help in automatically performing the transformations required prior to the application of the fixed point methods.